Starling chords: Difference between revisions

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'''Starling chords''' are [[Dyadic chord#Essentially tempered dyadic chords|essentially tempered dyadic chords]] tempered by the ~~starling comma,~~ [[~~126/125]]. The~~ starling ~~[[tetrad~~]] ~~is an 126/125-tempered~~ [[~~chord~~]] ~~which in close position (~~[[~~octave reduction|reduced to an octave]]) consists of a chain of three approximate [[6/5]] minor thirds and an approximate [[7/6]] subminor third, in a tuning which tempers out the starling comma,~~ 126/125, and hence causes it to close on the octave; that is, it is a starling-tempered 6/5-6/5-6/5-7/6, which may also be regarded as a tempered 1-6/5-10/7-12/7; it is also known as the septimal semicomma diminished seventh chord. In a meantone tuning the starling tetrad is three minor thirds and an augmented second, which is the traditional diminished seventh chord of common practice music in the meantone era. Also familiar from the music of common practice is the diminished [[triad]]~~, a tempering of 1-6/5-10/7 with steps 6/5-6/5-7/5 when understood septimally~~.

'''Starling chords''' are [[Dyadic chord#Essentially tempered dyadic chords|essentially tempered dyadic chords]] tempered by the [[Starling family|starling]] [[comma]], [[126/125]].

~~Other essentially~~ tempered starling [[triad]]s are the 1-5/4-7/5 chord with steps 5/4-10/9-10/7 and its inversion, 1-5/4-9/5 with steps 5/4-10/7-10/9. For starling tetrads, we also have the 1-5/4-3/2-9/5 chord with steps 5/4-6/5-6/5-10/9 and its inversion, 1-6/5-3/2-5/3; the 1-5/4-7/5-7/4 chord with steps of 5/4-10/9-5/4-8/7; the 1-5/4-7/5-9/5 chord with steps of 5/4-10/9-9/7-10/9; and the 1-9/8-5/4-9/5 chord with steps of 9/8-10/9-10/7-10/9.

The starling [[tetrad]] is an 126/125-tempered [[chord]] which in close position ([[octave reduction|reduced to an octave]]) consists of a chain of three approximate [[6/5]] minor thirds and an approximate [[7/6]] subminor third, in a tuning which [[Tempering out|tempers out]] the starling comma, 126/125, and hence causes it to close on the [[octave]]. That is, it is a starling-tempered 6/5-6/5-6/5-7/6, which may also be regarded as a tempered 1-6/5-10/7-12/7; it is also known as the septimal semicomma diminished seventh chord. In a [[meantone]] tuning the starling tetrad is three minor thirds and an augmented second, which is the traditional diminished seventh chord of common practice music in the meantone era.

Also familiar from the music of common practice is the diminished [[triad]], a tempering of 1-6/5-10/7 with steps 6/5-6/5-7/5 when understood [[7-limit|septimally]]. Other essentially tempered starling triads are the 1-5/4-7/5 chord with steps 5/4-10/9-10/7 and its inversion, 1-5/4-9/5 with steps 5/4-10/7-10/9.

For starling tetrads, we also have the 1-5/4-3/2-9/5 chord with steps 5/4-6/5-6/5-10/9 and its inversion, 1-6/5-3/2-5/3; the 1-5/4-7/5-7/4 chord with steps of 5/4-10/9-5/4-8/7; the 1-5/4-7/5-9/5 chord with steps of 5/4-10/9-9/7-10/9; and the 1-9/8-5/4-9/5 chord with steps of 9/8-10/9-10/7-10/9.

There is also a unique starling [[pentad]], 1-9/8-5/4-3/2-9/5, with steps of 9/8-10/9-6/5-6/5-10/9, a modified complete ninth chord. In the transposition 1-6/5-4/3-3/2-5/3 with steps 6/5-10/9-9/8-10/9-6/5 we can more easily see that the steps are symmetrical around 9/8.

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Starling chords come to triads: 3, tetrads: 6, pentads: 1, for a total of 10.

Equal temperaments with starling chords include 19, 27, 31, 43, 46, 50, 58, 77, 185 and 265c, and using the patent val tuning, 34 and 65.

[[EDO|Equal temperaments]] with starling chords include [[19edo|19]], [[27edo|27]], [[31edo|31]], [[43edo|43]], [[46edo|46]], [[50edo|50]], [[58edo|58]], [[77edo|77]], [[185edo|185]] and [[265edo|265c]], and using the [[patent val]] tuning, [[34edo|34]] and [[65edo|65]].

[[Category:7-odd-limit]]

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[[Category:Tetrad]]

[[Category:Pentad]]

~~[[Category:Todo:add links]]~~

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-'''Starling chords''' are [[Dyadic chord#Essentially tempered dyadic chords|essentially tempered dyadic chords]] tempered by the starling comma, [[126/125]]. The starling [[tetrad]] is an 126/125-tempered [[chord]] which in close position ([[octave reduction|reduced to an octave]]) consists of a chain of three approximate [[6/5]] minor thirds and an approximate [[7/6]] subminor third, in a tuning which tempers out the starling comma, 126/125, and hence causes it to close on the octave; that is, it is a starling-tempered 6/5-6/5-6/5-7/6, which may also be regarded as a tempered 1-6/5-10/7-12/7; it is also known as the septimal semicomma diminished seventh chord. In a meantone tuning the starling tetrad is three minor thirds and an augmented second, which is the traditional diminished seventh chord of common practice music in the meantone era. Also familiar from the music of common practice is the diminished [[triad]], a tempering of 1-6/5-10/7 with steps 6/5-6/5-7/5 when understood septimally.
+'''Starling chords''' are [[Dyadic chord#Essentially tempered dyadic chords|essentially tempered dyadic chords]] tempered by the [[Starling family|starling]] [[comma]], [[126/125]].
-Other essentially tempered starling [[triad]]s are the 1-5/4-7/5 chord with steps 5/4-10/9-10/7 and its inversion, 1-5/4-9/5 with steps 5/4-10/7-10/9. For starling tetrads, we also have the 1-5/4-3/2-9/5 chord with steps 5/4-6/5-6/5-10/9 and its inversion, 1-6/5-3/2-5/3; the 1-5/4-7/5-7/4 chord with steps of 5/4-10/9-5/4-8/7; the 1-5/4-7/5-9/5 chord with steps of 5/4-10/9-9/7-10/9; and the 1-9/8-5/4-9/5 chord with steps of 9/8-10/9-10/7-10/9.
+The starling [[tetrad]] is an 126/125-tempered [[chord]] which in close position ([[octave reduction|reduced to an octave]]) consists of a chain of three approximate [[6/5]] minor thirds and an approximate [[7/6]] subminor third, in a tuning which [[Tempering out|tempers out]] the starling comma, 126/125, and hence causes it to close on the [[octave]]. That is, it is a starling-tempered 6/5-6/5-6/5-7/6, which may also be regarded as a tempered 1-6/5-10/7-12/7; it is also known as the septimal semicomma diminished seventh chord. In a [[meantone]] tuning the starling tetrad is three minor thirds and an augmented second, which is the traditional diminished seventh chord of common practice music in the meantone era.
+Also familiar from the music of common practice is the diminished [[triad]], a tempering of 1-6/5-10/7 with steps 6/5-6/5-7/5 when understood [[7-limit|septimally]]. Other essentially tempered starling triads are the 1-5/4-7/5 chord with steps 5/4-10/9-10/7 and its inversion, 1-5/4-9/5 with steps 5/4-10/7-10/9.
+For starling tetrads, we also have the 1-5/4-3/2-9/5 chord with steps 5/4-6/5-6/5-10/9 and its inversion, 1-6/5-3/2-5/3; the 1-5/4-7/5-7/4 chord with steps of 5/4-10/9-5/4-8/7; the 1-5/4-7/5-9/5 chord with steps of 5/4-10/9-9/7-10/9; and the 1-9/8-5/4-9/5 chord with steps of 9/8-10/9-10/7-10/9.
 There is also a unique starling [[pentad]], 1-9/8-5/4-3/2-9/5, with steps of 9/8-10/9-6/5-6/5-10/9, a modified complete ninth chord. In the transposition 1-6/5-4/3-3/2-5/3 with steps 6/5-10/9-9/8-10/9-6/5 we can more easily see that the steps are symmetrical around 9/8.
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 Starling chords come to triads: 3, tetrads: 6, pentads: 1, for a total of 10.
-Equal temperaments with starling chords include 19, 27, 31, 43, 46, 50, 58, 77, 185 and 265c, and using the patent val tuning, 34 and 65.
+[[EDO|Equal temperaments]] with starling chords include [[19edo|19]], [[27edo|27]], [[31edo|31]], [[43edo|43]], [[46edo|46]], [[50edo|50]], [[58edo|58]], [[77edo|77]], [[185edo|185]] and [[265edo|265c]], and using the [[patent val]] tuning, [[34edo|34]] and [[65edo|65]].
 [[Category:7-odd-limit]]
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 [[Category:Tetrad]]
 [[Category:Pentad]]
-[[Category:Todo:add links]]